Making a laminate twist

The properties of fiber-reinforced composite structures can be tailored with layer orientations and stacking sequence. For example, you can design your laminate so that under a tensile load it twists, but at the same time it does not bend nor shear. Typically, distortions are unwanted but sometimes they may be beneficial when properly understood and utilized.

The Classical Lamination Theory (CLT) can be used for the analysis and design of relatively thin and flat laminated structures. In CLT the relation between the forces and deformations are defined with [A], [B] and [D] matrices (see Eq.1), where [A] is the in-plane, [B] the coupling, and [D] the flexural stiffness matrix of the laminate. Typically, loads are represented in the form of resultant forces {N} and moments {M} per unit width. Deformations are presented in the form of mid-plane strains {ε0} and laminate curvatures {κ}, respectively.

CLT constitutive relations

Typically one needs to solve the deformations due to the applied loads rather than vice versa. Therefore, the above constitutive relation is rewritten in the form presented in Eq.2, where [a], [b] and [d] are the in-plane, coupling, and flexural compliance matrices of the laminate. The combined compliance matrix is the inverse of the 6 by 6 stiffness matrix described in Eq.3. Now the constitutive equations are written in the expanded matrix format:

{eps} = [a]{N}

Here we have defined that the laminate is loaded only in the axial direction. Since no other load components are involved in this example study, we can basically neglect the last five columns of the matrix. The a11 term directly reveals how much the laminate extends in the direction of the load. Respectively, the a12 term indicates how much the laminate contracts in the transverse direction due to the axial load, and so on. With layer orientations and stacking sequence one can influence which elements of the compliance matrix are nonzero and also their magnitude and, therefore, control the deformation of the laminate. Let’s have a more thorough look on four laminate lay-ups and respective [a] and [b] matrices.

A [-45/+45]s laminate is symmetric and therefore the coupling compliance matrix [b] is zero. Consequently, the laminate extension and bending are uncoupled. The laminate is equally stiff in the transverse direction and therefore a22=a11. In balanced laminates for each off-axis layer with a positive orientation angle, there is an identical layer oriented in a negative angle of the same magnitude. The example laminate is also balanced. For balanced laminates shear and extension are uncoupled, which is indicated by a13=a23=0.

CLT coupling

A [45] laminate is symmetric as well. For symmetric laminates there exists an identical layer with the same orientation placed symmetrically about the midplane of the laminate. Laminates with odd number of plies can be symmetric as well since the symmetry plane maybe in the middle of the middle layer. The off-axis ply without a balancing layer results in a nonzero term a13, which is the cause for the shear deformation under an axial load.

A [0/90] laminate is a cross-ply laminate in which all layers are oriented in the longitudinal or transverse direction. For a cross-ply laminate, in-plane shear and extension are uncoupled (a13=0). Cross-ply laminates can be symmetric, antisymmetric or unsymmetric. Antisymmetric and unsymmetric laminates have nonzero terms in the [b] matrix. Common for all cross-ply laminates is that they do not twist (b13=0) under an axial load. The example laminate is antisymmetric and the uniaxial load generates cylindrical bending only (b11≠0, b12=0), i.e. bending takes place only around the y-axis. Antisymmetric laminates are laminates in which for each off-axis layer with positive orientation, there exists an identical layer placed symmetrically with respect to the midplane of the laminate and oriented in a negative angle of the same magnitude. The antisymmetric pair of the longitudinal ply is the transversal ply.

A [-45/+45] laminate is antisymmetric as well ([b]≠0). In the plane of the laminate the antisymmetric off-axis layers compensate each other and the shear deformation do not exist under an axial load (a13=0). Instead, in the coupling matrix the same antisymmetric pairs amplify each other (b13≠0) and make the laminate to twist. The coupling terms related to the bending are compensating each other (b11=b12=0).

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